PCM4: Cichlid Bayesian Linear And Phylogenetic Mixed Models

PCM4 introduces posterior inference and phylogenetic random effects through a cichlid parasite study. It connects observation-level records to a species-level analysis, compares a Gaussian baseline with Bayesian fits, and uses the inverse phylogenetic covariance in an MCMCglmm model.

flowchart LR
    observations["Cichlid parasite<br/>observations"]
    species["Species-level<br/>summary"]
    baseline["Gaussian baseline"]
    bayes["Bayesian linear model"]
    inverse["Inverse phylogenetic<br/>covariance"]
    mixed["Phylogenetic<br/>mixed model"]
    compare["Posterior diagnostics<br/>and PGLS comparison"]

    observations --> species --> baseline --> bayes
    species --> inverse --> mixed
    bayes --> compare
    mixed --> compare

The Posterior Question In One View

Dimension	PCM4 contract
scientific object	a species-level cichlid parasite response analyzed by Gaussian GLS, Bayesian Gaussian chains, a phylogenetic mixed model, and a Brownian PGLS comparator
central idea	covariance-model regression and phylogenetic random effects can describe related dependence while producing different estimands, uncertainty objects, and diagnostic obligations
evidence role	defines the population, inverse-covariance, prior, chain, fixed-effect, variance-component, and comparator identities required for posterior correspondence
present conclusion	all seven manifests are not_comparable, use bundle_contract_only, and report governed_primary_output_count: 0; no governed posterior is available
relation to PCM5	establishes the Gaussian posterior and random-effect denominator that PCM5 extends to non-Gaussian families, repeated records, and derived quantities
forbidden inference	adjacent native Bayesian or PGLS capability does not reproduce the source MCMCglmm chains, posterior summaries, or variance components

PCM4 preserves the exact analytical bridge that is still missing. The dossier can support a statement about source identity and closure requirements; it cannot support a coefficient, credible interval, convergence, or variance correspondence claim.

Scientific Questions

• Does the parasite response retain the baseline predictor relationship after posterior uncertainty is represented?
• How much variation is assigned to the phylogenetic random effect?
• Are fixed-effect estimates and uncertainty coherent with the Brownian PGLS comparator?
• Do chain diagnostics support interpretation of posterior summaries?

The inverse covariance is not an implementation detail: taxon order, rootedness, branch lengths, and alignment with the species table determine the random-effect structure.

What The Phylogenetic Random Effect Represents

The mixed model assigns each species a latent deviation whose joint covariance follows the tree. Closely related cichlids are allowed to have more similar deviations, while residual variance represents variation not assigned to that tree-linked component under the model. The posterior estimates both components with the fixed effects rather than applying a correction after fitting.

This is related to Brownian PGLS but not identical to it. PGLS places the tree-derived structure in the residual covariance; the mixed model represents it as a random effect with its own variance posterior. Similar fixed-effect estimates can be informative, but they do not make the likelihoods, variance components, or uncertainty objects interchangeable.

Why PCM4 Is More Than Bayesian PGLS

PCM4 deliberately preserves four related estimators because each changes a different part of the scientific contract.

Estimator	Dependence model	Uncertainty object	Question it answers
Gaussian GLS baseline	independent residuals	maximum-likelihood sampling uncertainty	what fixed relationship appears without phylogenetic covariance?
Bayesian Gaussian model	independent residuals with declared priors	posterior draws for fixed and residual terms	how do prior and posterior estimation change the same mean structure?
phylogenetic mixed model	phylogenetic random effect plus residual variation	joint posterior for fixed effects and variance components	how much modeled variation is assigned to the tree-linked random effect?
Brownian PGLS comparator	residual covariance derived directly from the tree	maximum-likelihood coefficient and fit uncertainty	how does a covariance-model treatment compare with the random-effect treatment?

Close fixed-effect point estimates would not make these estimators interchangeable. A comparison must preserve priors, variance components, posterior dependence, chain diagnostics, covariance scaling, and the different meaning of PGLS versus random-effect uncertainty.

Model Components

The central phylogenetic mixed model can be read as:

y = X beta + Z u_phylogeny + epsilon
u_phylogeny ~ Normal(0, sigma_phylogeny^2 A)
epsilon     ~ Normal(0, sigma_residual^2 I)

A is the tree-derived covariance in the aligned species order. The source workflow supplies its inverse to MCMCglmm. A valid comparison must therefore align not only fixed-effect formulas but also covariance scaling, random-effect levels, priors on variance components, observation aggregation, and posterior summaries.

Random Effects And Residual Covariance Are Related, Not Identical Records

A Brownian PGLS fit and a phylogenetic mixed model can encode related tree structure while assigning it to different model components. Similar fixed effects are therefore a cross-estimator diagnostic, not proof that the models share one likelihood or uncertainty object.

Identity	Brownian PGLS	Phylogenetic mixed model
phylogenetic dependence	residual covariance is structured directly by the tree	a latent random effect has tree-derived covariance
residual term	part of the combined fitted covariance contract	separately parameterized residual variance
principal uncertainty	maximum-likelihood sampling uncertainty	joint posterior over fixed effects and variance components
tree scaling consequence	changes residual weighting and likelihood	changes random-effect covariance and variance interpretation
reported comparison	coefficients, fit, covariance and diagnostics	fixed posterior, phylogenetic/residual variance posteriors, chains and diagnostics

To compare their scientific conclusions, retain whether effects are marginal or conditional, how covariance is scaled, and which variance components are integrated or estimated. To compare implementations, first reproduce each estimator under its own identity; do not use one as the missing output of the other.

The Preserved Source Analysis

The preserved Lund program is a connected analysis, not a collection of interchangeable model calls. These source objects define the handoffs that a reproduction must retain:

Source object	Computation	Evidence responsibility
fish	observation-level cichlid.csv records	preserves repeated measurements before aggregation
fishtree	cichlidtree.nex, pruned to represented taxa	fixes topology, branch lengths, and the eligible taxon set
fishSum	records grouped by treetip, species, and mating_system	fixes the species-level population used by the Gaussian analyses
glsFish	maximum-likelihood GLS for log(parasite_abundance + 1) ~ gut_length	supplies the non-phylogenetic Gaussian baseline
m1	Gaussian MCMCglmm with the same fixed formula	introduces posterior estimation without a phylogenetic random effect
m1a, m1b, m1c	independently invoked Gaussian chains	supply the between-chain diagnostic population
inv.fishphylo	inverseA(..., nodes = "TIPS", scale = TRUE)	supplies the ordered, scaled inverse covariance
m2	Gaussian MCMCglmm with random effect treetip	estimates fixed effects and phylogenetic and residual variance
pgls6	Brownian GLS with corBrownian(..., form = ~treetip)	supplies a maximum-likelihood phylogenetic comparator

flowchart LR
    fish["fish<br/>observation records"]
    tree["fishtree<br/>pruned tree"]
    summary["fishSum<br/>species population"]
    gls["glsFish<br/>Gaussian baseline"]
    chains["m1 / m1a–m1c<br/>posterior and diagnostics"]
    inverse["inv.fishphylo<br/>scaled inverse covariance"]
    mixed["m2<br/>phylogenetic mixed model"]
    pgls["pgls6<br/>Brownian GLS"]

    fish --> summary
    tree --> summary
    summary --> gls
    summary --> chains
    tree --> inverse
    summary --> mixed
    inverse --> mixed
    summary --> pgls
    tree --> pgls

Matching only the final coefficient discards this structure. A governed comparison must show that each output was computed from the same analytical population and the same tree identity before it compares numerical values.

Evidence Decomposition

Bundle	Evaluated surface	Manifest verdict
evidence-001	data–tree alignment and species-level summarization	not_comparable
evidence-002	non-phylogenetic Gaussian baseline	not_comparable
evidence-003	Bayesian Gaussian chain diagnostics and summaries	not_comparable
evidence-004	inverse-tree construction and random-effect setup	not_comparable
evidence-005	phylogenetic mixed-model fixed-effect posterior	not_comparable
evidence-006	phylogenetic variance-component posterior	not_comparable
evidence-007	Brownian PGLS comparison	not_comparable

All seven bundles are governed review units. Their comparison verdicts are not_comparable because no governed comparison lane currently executes and adjudicates the claim-specific reference and Bijux computations. The locally available PGLS comparator is useful adjacent capability, but it is not a replacement posterior.

Keep The Four Estimation Questions Separate

The source analysis contains four related estimators. Similar coefficient names do not make their likelihoods, uncertainty, or covariance structures interchangeable.

Estimator	Scientific role	Required comparison identity	Invalid substitution
glsFish	non-phylogenetic Gaussian maximum-likelihood baseline	species population, response transform, formula, variance and likelihood convention	a successful fit does not establish posterior behavior
m1 and independent chains	non-phylogenetic Gaussian posterior	priors, proposal/chain policy, monitored parameters, draw and diagnostic population	a baseline GLS standard error is not a credible interval
m2	phylogenetic Gaussian mixed-model posterior	all m1 identity plus inverse covariance, treetip levels, scaling and variance components	PGLS covariance does not reproduce random-effect posterior draws
pgls6	Brownian maximum-likelihood phylogenetic comparator	tree, taxon order, Brownian covariance, response/formula and ML convention	agreement cannot promote m2 fixed or variance posteriors

These questions can form a coherent triangulation only when each estimator has its own governed execution and result. A favorable baseline or PGLS result may diagnose a future posterior difference; it cannot inherit the missing MCMCglmm computation or verdict.

For a cross-estimator conclusion, compare the estimand first: coefficient scale, conditional or marginal interpretation, covariance treatment, and uncertainty object. Then retain each result and diagnostic separately before asking whether their bounded conclusions agree.

Retained Record State

Record layer	Present in every bundle	What it establishes
claim and manifest	yes	claim identity, direct-parity intent, source basis, ownership, limitation, and not_comparable verdict
authored reference.R	yes	writes a reference-contract record pointing to the preserved Lund source; it does not execute MCMCglmm
authored analysis.py	yes	writes a bundle_contract_only run record; BUILD_SCRIPT is None and PRIMARY_OUTPUTS is empty
checks and provenance	yes	required files, source locators, expected verdict, and freshness dependencies
reviewer summaries	yes	human and JSON projections of the manifest and claim record
governed primary outputs	no	every results manifest records governed_primary_output_count: 0
claim-level comparison observations	no	there are no aligned coefficients, posterior draws, diagnostics, tolerances, or adjudicated mismatch rows

The authored programs are real and executable contract wrappers, but they are not yet scientific reproductions. The results manifests name reference-contract.json and analysis-run.json as local rerun products; those ephemeral records are not governed primary outputs and are not retained in the checked-in results directories.

Verify The Present Record Without Inferring A Posterior

A reader can audit PCM4 today without pretending that the missing posterior comparison ran. Read each bundle in this order:

1. use the manifest and claim record to identify the bounded statement and current not_comparable verdict;
2. use the input manifest, dataset registry, and provenance to establish which source program, cichlid records, tree, and workspace objects are named;
3. inspect reference.R and analysis.py to confirm that they emit contract records rather than MCMCglmm and Bijux scientific outputs;
4. inspect checks.json and the results manifest to confirm that governed primary-output count is zero;
5. use the closure contract to identify the missing execution, posterior, diagnostic, and observation records.

This traversal supports a precise present-tense statement: PCM4 preserves the source, input identities, model decomposition, and review contract. It does not support a coefficient, credible interval, variance component, convergence finding, or Brownian-comparator correspondence result.

Current Verdict

All seven bundle manifests record not_comparable. This is the correct status for the present record: source and data provenance are governed, and nearby Gaussian and PGLS surfaces can be inspected, but the bundles retain zero governed primary outputs and no repository-owned lane currently executes the reference posterior and compares aligned samples or summaries.

The status must remain separate from data preparation quality. A verified tree–table join or Brownian PGLS coefficient can support its own bounded claim without causing fixed-effect, variance-component, or convergence claims from the mixed model to match.

Why Native Bayesian Capability Does Not Close PCM4

The runtime owns native Bayesian surfaces for several DNA, clock, continuous trait, discrete trait, posterior-diagnostic, and posterior-predictive contracts. PCM4 asks a different question: a Gaussian hierarchical model with the lecture's fixed formula, a treetip random effect, the scaled inverse phylogenetic covariance, MCMCglmm priors, and comparable fixed-effect and variance-component posteriors.

Reusing a sampler, effective-sample-size calculation, Brownian trait model, or PGLS covariance does not create that hierarchical estimand. A PCM4 comparison needs an explicitly owned mixed-model state and execution contract or a governed adapter that preserves the external computation. Until then, nearby Bayesian code is relevant engineering context, not PCM4 primary evidence.

Verdict And Follow-Through

Axis	Current state	Consequence
bundle verdict	all seven are not_comparable	no PCM4 bundle currently establishes posterior correspondence
source and input identity	governed	reviewers can identify the lecture, data, tree, workspace objects, and intended analysis; this does not prove a successful analytical rerun
local adjacent capability	Gaussian and Brownian PGLS surfaces exist	these methods can support their own claims but cannot inherit MCMCglmm posterior claims
resolution work	open	a matched-prior, matched-parameterization posterior lane and adjudicated observations are still required

The open item is a workflow state, not a sixth verdict value. Retain the not_comparable verdict until the missing execution and comparison record exists; do not rewrite it as “open,” “pending,” or “partially matched.”

What Would Make The Posterior Comparable

A comparison must first replace contract-only wrappers with governed reference and Bijux executions. It must identify the exact source formula, family, link, fixed and random terms, tree inverse, taxon order, covariance scaling, priors, chain policy, monitored quantities, and summary transformations. It must execute the reference and comparison surfaces, retain raw and normalized posterior observations, define tolerances or distributional rules, and account for every failed or excluded chain and parameter. Source reconstruction alone does not satisfy this contract.

Minimum Numerical Denominator

Every governed run must report the number of initialized chains, completed chains, retained draws per chain, monitored parameters, and eligible comparison rows. The source schedule uses 110,000 iterations, a burn-in of 10,000, and thinning by 100, but those arguments do not prove how many rows were emitted or usable. The retained artifacts must establish the actual denominator and make failed chains, absent parameters, non-finite draws, and exclusions explicit.

Posterior summaries then need a stable parameter key, chain identity, draw or summary identity, scale, interval convention, and source model. That record is what permits a reviewer to distinguish missing computation from numerical disagreement.

Inputs And Provenance

The study carries cichlid.csv, cichlidtree.nex, and fishdata.RData with a dataset catalog, source provenance, and runtime/reference mapping.

Posterior Review Contract

A usable posterior comparison needs the prior, family, link, chain length, burn-in, thinning, seed policy, effective sample size, autocorrelation, and convergence evidence. Point estimates alone cannot establish parity. Fixed effects, phylogenetic variance, residual variance, and their credible intervals must be compared under matching parameterization.

At least two independently initialized chains or an explicitly justified alternative are needed to assess between-chain behavior. Effective sample size and autocorrelation qualify each monitored quantity; a large retained row count does not by itself demonstrate effective posterior information.

Review Priorities

• Reconcile observation species with tree tips before aggregation.
• Record exactly how repeated records become a species-level table.
• Verify inverse-covariance taxon order and scaling.
• Keep the Gaussian baseline, Bayesian non-phylogenetic fit, and phylogenetic mixed model as distinct comparisons.
• Preserve posterior diagnostics with every reported summary.
• State that MCMCglmm remains the posterior computation owner.

See the mixed-model evidence boundary for the current closure requirements. PCM5 extends the same system to count, binary-style, and repeated-measure models.